In a constant acceleration motion, Is it possible to have constant speed?

[MatinSAR]

Homework Statement

In a constant acceleration motion, Is it possible to have constant speed?

Relevant Equations

Please see below.

I think it's not possible.
In 3D for constant acceleration we have : $\vec v = \vec v_0 + \vec a t$

It's a line in 3 dimension so velocity's magnitude(speed) is changing with time.
I appreciate any better idea.

 

[kuruman]

You are correct, it is not possible. The closest you can come is constant speed and constant magnitude of the acceleration which is the case of uniform circular motion.

 

[Ibix]

$\vec a=0$?

 

[MatinSAR]

$\vec a=0$?

Then we have no acceleration.

 

[Ibix]

Then we have no acceleration.

Zero is a constant. If you mean "constant non-zero acceleration" I would say you should say so.

 

[kuruman]

Then we have no acceleration.

That is true but $0$ is a perfectly good constant.

 

[MatinSAR]

Zero is a constant. If you mean "constant non-zero acceleration" I would say you should say so.

You are right. I meant "constant non-zero acceleration". Thanks for your reply @Ibix .

That is true but $0$ is a perfectly good constant.

Thank you for your time @kuruman .

 

[haruspex]

It's a line in 3 dimension

Not sure what you mean. The trajectory from that equation does not have to be a straight line.
But how about a proof?
Constant speed implies $\vec v\cdot\vec v=c$. Differentiating, $\vec v.{\vec a}=0=\vec v_0\cdot\vec a+a^2t$. That can only be true for varying t and constant $\vec a$ if a=0.

 

[MatinSAR]

Not sure what you mean. The trajectory from that equation does not have to be a straight line.

Why? Isn't it similar to $\vec r=\vec r_0 + s\vec t$ which is equation of a line in vector form?

But how about a proof?
Constant speed implies $\vec v\cdot\vec v=c$. Differentiating, $\vec v.{\vec a}=0=\vec v_0\cdot\vec a+a^2t$. That can only be true for varying t and constant $\vec a$ if a=0.

Far better idea!
Thanks a lot for your time.

 

[haruspex]

Why? Isn't it similar to $\vec r=\vec r_0 + s\vec t$ which is equation of a line in vector form?

Consider $\vec a$ normal to $\vec v_0$.

 

[MatinSAR]

Consider $\vec a$ normal to $\vec v_0$.

Good point, Thanks! I didn't think about it.

 

[Mister T]

The equations used to describe motion with constant acceleration include, as a possibility, a constant acceleration of zero.